How to Draw a Triangle on a Coordinate Plane

Scaling a Triangle in the Coordinate Plane

Alignments to Content Standards: Thousand-GPE.B.6

Job

Below is a picture of $\triangle ABC$ with vertices lying on filigree points:

1. Draw the image of $\triangle ABC$ when information technology is scaled with a scale factor of $\frac{i}{iii}$ about the vertex $A$: label this triangle $A^\prime B^\prime C^\prime$ and find the coordinates of the points $A^\prime$, $B^\prime number$, and $C^\prime number$.
2. Draw the image of $\triangle ABC$ when it is scaled with a scale factor of $\frac{ii}{3}$ virtually the vertex $B$: label this triangle $A^{\prime \prime}B^{\prime \prime number}C^{\prime \prime number}$ and observe the coordinates of the points $A^{\prime number \prime}$, $B^{\prime number \prime number}$, and $C^{\prime \prime}$.
3. How does $A^{\prime \prime}$ compare $B^{\prime}$? Why?

IM Commentary

The goal of this task is to employ dilations to a triangle in the coordinate airplane. Students volition need to find the coordinates of a point which cuts a given segment into 1 third (part a) and into 2 thirds (function b). Because the centers for the two dilations are vertices of $\triangle ABC$ and the sum of the dilation factors (1 third and two thirds) is one, the two dilated triangles share a vertex. A more than challenging question, in place of (b) and (c), would be to inquire students which dilations they tin can apply almost $B$ and so that the epitome will share a vertex with the triangle synthetic in part (a): in that location are two possibilities, namely $\frac{2}{3}$ and 1.

Solution

1. Since $A$ is the center of the dilation, it does non motion and we will have $A^\prime number = A$. For vertex $B$, we know that $B^\prime$ volition lie on $\overleftrightarrow{AB}$ considering dilations preserve lines through the centre of dilation. Since the calibration factor is $\frac{ane}{3}$ we know that $|AB^\prime number| = \frac{one}{3}|AB|$. From $A$ to $B$ is 4 units to the right and one unit up. Therefore, one third of the way from $A$ to $B$ (along $\overline{AB}$) volition exist the signal $B^\prime = \left(1+ \frac{4}{three}, 1 + \frac{1}{iii}\correct)$. Applying this technique to $C$ which is 2 units to the correct and 4 units up from $A$, we discover $$ C^\prime = \left(1\frac{ii}{three}, ii\frac{1}{3} \correct). $$

   The scaled triangle $A^\prime number B^\prime C^\prime$ is pictured below:

   

2. Since $B$ is the middle of the dilation, information technology does not motion and nosotros will take $B^{\prime number \prime} = B$. For vertex $A$, we know that $A^{\prime \prime}$ will lie on $\overleftrightarrow{AB}$ because dilations preserve lines through the center of dilation. Since the scale cistron is $\frac{2}{three}$ we know that $|BA^{\prime \prime}| = \frac{2}{three}|BA|$. From $B$ to $A$ is 4 units to the left and ane unit downward. Therefore, two thirds of the manner from $B$ to $A$ (along $\overline{AB}$) will be the point $A^{\prime \prime} = \left(5 - \frac{viii}{3}, 2 - \frac{two}{3}\correct)$. Applying this technique to $C$ which is 2 units to the left and iii units up from $B$, we find $$ C^{\prime \prime number} = \left(iii\frac{2}{3}, 4 \right). $$

   The scaled triangle $A^{\prime number \prime} B^{\prime \prime} C^{\prime \prime number}$ is pictured below:

   

3. The calculations above testify that $B^\prime number = A^{\prime \prime} = \left( 2 \frac{i}{3}, i \frac{i}{3} \right)$. To see why these vertices are the aforementioned, note that the sum of the dilation factors $\frac{1}{3}$ and $\frac{2}{3}$ is 1. The dilation nearly $A$ maps $\overline{AB}$ to a the first third of $\overline{AB}$ while the dilation about $B$ maps $\overline{AB}$ to the 2d two thirds of this segment. The two scaled triangles with a shared vertex are pictured below:

   

Scaling a Triangle in the Coordinate Plane

Below is a picture of $\triangle ABC$ with vertices lying on filigree points:

1. Depict the paradigm of $\triangle ABC$ when it is scaled with a scale factor of $\frac{1}{3}$ nearly the vertex $A$: label this triangle $A^\prime B^\prime number C^\prime$ and notice the coordinates of the points $A^\prime number$, $B^\prime$, and $C^\prime$.
2. Draw the prototype of $\triangle ABC$ when it is scaled with a scale factor of $\frac{2}{3}$ nearly the vertex $B$: label this triangle $A^{\prime number \prime}B^{\prime number \prime}C^{\prime number \prime}$ and detect the coordinates of the points $A^{\prime \prime number}$, $B^{\prime \prime number}$, and $C^{\prime number \prime}$.
3. How does $A^{\prime number \prime number}$ compare $B^{\prime number}$? Why?

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Source: https://tasks.illustrativemathematics.org/content-standards/HSG/GPE/B/6/tasks/1867

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